HYPERSPECTRAL IMAGE KERNEL SPARSE SUBSPACE CLUSTERING WITH SPATIAL MAX POOLING OPERATION
Hongyan Zhang
a,b
, Han Zhai
b
, Wenzhi Liao
a
, Liqin Cao
c
, Liangpei Zhang
b
, Aleksandra Pižurica
a a
Ghent University, Dept. Telecommunications and Information Processing, TELIN-IPI-iMinds, Ghent, Belgium
b
The State Key Laboratory of Information Engineering in Surveying, Mapping, and Remote Sensing, Wuhan University, China
c
School of Printing and Packaging, Wuhan University, China zhanghongyan, zhaihan, clq, zlp62whu.edu.cn, wenzhi.liao, Aleksandra.Pizuricatelin.ugent.be
ICWG IIIVII KEY WORDS:
Hyperspectral image, nonlinear processing, spatial max pooling, SSC, kernel
ABSTRACT: In this paper, we present a kernel sparse subspace clustering with spatial max pooling operation KSSC-SMP algorithm for
hyperspectral remote sensing imagery. Firstly, the feature points are mapped from the original space into a higher dimensional space with a kernel strategy. In particular, the sparse subspace clustering SSC model is extended to nonlinear manifolds, which can better
explore the complex nonlinear structure of hyperspectral images HSIs and obtain a much more accurate representation coefficient matrix. Secondly, through the spatial max pooling operation, the spatial contextual information is integrated to obtain a smoother
clustering result. Through experiments, it is verified that the KSSC-SMP algorithm is a competitive clustering method for HSIs and outperforms the state-of-the-art clustering methods.
1. INTRODUCTION
Hyperspectral sensors can acquire nearly continuous spectral bands with hundreds of channels to capture the diagnostic
information of land-cover materialsZhang, et. al., 2014a, which opens up new possibilities for remote sensing applications, such
as mineral exploration, fine agriculture, disaster monitoring, and so on Landgrebe, 2002, Zhang, et. al., 2014b . As an
unsupervised information extraction technique, clustering is a basic tool of hyperspectral image HSI applications. However,
due to the complex nonlinear structure and large spectral variability, clustering HSIs is still a very challenging task.
The traditional HSI clustering methods, such as k-means Lloyd, 1982 and fuzzy c-means FCM Bezdek, 2013, attempt to
segment pixels using only spectral measurements. Unfortunately, such methods often fail to achieve satisfactory clustering results
because of the limited discriminative capability. Therefore, in recent years, researchers have begun to develop spectral-spatial
clustering methods which consider spectral measurements together with spatial information to improve the clustering
performance, such as FCM_S1 Chen, and Zhang, 2004, and k- means_S Luo, et al., 2003. However, these methods still have
limited clustering performance due to large spectral variability of HSIs.
The sparse subspace clustering SSC algorithm was recently proposed Elhamifar, and Vidar, 2013, and has achieved great
success in the face recognition and motion segmentation fields. Based on the assumption that pixels with different spectra from
one land cover class lie in the same subspace, the SSC algorithm shows great potential in HSI clustering. Based on the subspace
model, the spectral variability problem can be effectively relieved Zhang, et. al., 2016. Recently, a spectral-spatial sparse
subspace clustering algorithm was proposed to make full use of the spectral-spatial information of HSIs Zhang, et. al., 2016,
which significantly improves the clustering performance. However, the SSC model is based on the linear subspace model,
while HSIs are generally considered to be linearly inseparable. Therefore, these linear models cannot cope well with the
Corresponding author: Hongyan Zhang. inherently nonlinear structure of the HSIs.
In recent years, a number of approaches have been proposed to deal with the linearly inseparable obstacle in the classification
field. Among them, the kernel strategy is one of the most commonly used and effective methods. This approach maps the
HSI from the original feature space to a higher kernel feature space to make the problem linearly separable, which has been
successfully used in the kernel-based SVM classifier Mercier, and Lennon, 2003 . However, to the best of our knowledge, in
the HSI clustering domain, few clustering methods have been proposed to deal with the nonlinear structure of the HSIs up to
date. In view of this, in this paper, a novel kernel sparse subspace
clustering algorithm with spatial max pooling operation KSSC- SMP for hyperspectral remote sensing imagery was proposed,
which simultaneously explores the nonlinear structure and the inherent spectral-spatial attributes of HSIs. Firstly, we map the
feature points from the original feature space to a higher- dimensional space with the kernel strategy to make the feature
points linearly inseparable. And then, in order to fully exploit the spectral-spatial discrimination information of HSIs and the
potential of the SSC model, the spatial max pooling operation is introduced to incorporate the spatial information to improve the
clustering performance and guarantee spatial homogeneity of the clustering result.
2. SPARSE SUBSPACE CLUSTERING
In the SSC model, for an HSI with the size of M N p
, all the pixels can be seen as selected from a union of
l
affine subspaces
1 2
l
S S
S
of dimensions
1 l
i i
d
in the full space
p
with
1 2
l
d d
d p
, where
M
denotes the height of the image, N
represents the width of the image, and
p
is the number of band channels. By treating each pixel as a column vector, the HSI cube can be transformed into a 2-D
matrix
1 2
, ,
,
p MN MN
y y y
Y
. Then, with this 2-D- hyperspectral matrix itself being used as the representation
dictionary, the sparse coefficient matrix can be obtained by solving the following optimization problem:
This contribution has been peer-reviewed. doi:10.5194isprsarchives-XLI-B3-945-2016
945
min .
, ,
T
s t diag
C
C Y
YC N
C 0 C 1
1
1 where
MN MN
C
represents the representation coefficient matrix,
p MN
N
denotes the representation error matrix, and
MN
1 is
a vector whose elements are all ones. The condition
diag
C
is utilized to eliminate the trivial solution of each pixel being represented as a linear combination of itself Elhamifar, and
Vidar, 2013. The condition
T
C 1
1
means that it adopts the affine subspace model, which is a special linear subspace model.
As the - norm optimization problem is NP-hard, the relaxed
tractable
1
-norm is usually adopted:
1
min .
, ,
T
s t diag
C
C Y
YC N
C 0 C 1
1
2 The optimization problem in 2 can be effectively solved by the
alternating direction method of multipliers ADMM algorithm Mota, et. al., 2013. We then construct the similarity graph with
the obtained coefficient matrix in the symmetric form to strengthen connectivity of the graph Elhamifar and Vidar, 2013.
T
W
C C
3 The spectral clustering algorithm is then applied to the similarity
graph to obtain the final clustering result Ng, et. al., 2002 .
3. THE KERNEL SPARSE SUBSPACE CLUSTERING
